In the second half of the twentieth century Constantin Piron and Maria Pia Solèr proved that the number systems that a separable Hilbert space can use must be division rings. See: Baez, J. (2011) Division Algebras and Quantum Theory.
http://arxiv.org/abs/1101.5690 and Holland, S.S. (1995) Orthomodularity in Infinite Dimensions: A Theorem of M. Solèr. Bulletin of the American Mathematical Society, 32, 205-234
http://www.ams.org/journals/bull/1995-32-02/S0273-0979-1995-00593-8/

has been cited by the following article:

• TITLE: On the Origin of Electric Charge

  AUTHORS: Johannes Antonie Josephus van Leunen

  KEYWORDS: Quantum Physics, Mathematical Model, Electric Charge, Electric Field, Embedding Field, Photon, Wave Equation, Quaternionic Hilbert Space, Quaternionic Differential Calculus, Symmetry Flavor

  JOURNAL NAME: Open Access Library Journal, Vol.2 No.10, October 22, 2015

  ABSTRACT: By starting from a quaternionic separable Hilbert space as a base model, the paper uses the capabilities and the restrictions of this model in order to investigate the origins of the electric charge and the electric fields. Also, other discrete properties such as color charge and spin are considered. The paper exploits all known aspects of the quaternionic number system and it uses quaternionic differential calculus rather than Maxwell based differential calculus. The paper presents fields as mostly continuous quaternionic functions. The electric field is compared with another basic field that acts as a background embedding continuum. The behavior of photons is used in order to investigate the long range behavior of these fields. The paper produces an algorithm that calculates the electric charge of elementary particles from the symmetry properties of their local parameter spaces. The paper also shows that the usual interpretation of a photon as an electric wave is not correct.